Reproduction Number Research Articles

A multi-population approach to epidemiological modeling of Listeriosis transmission dynamics incorporating food and environmental contamination

Listeriosis is a food-borne disease that mainly affects pregnant women and newborns. We propose and analyze a deterministic model of Listeriosis by considering three groups of individuals: newborns, pregnant women, and others. Mathematical analysis of the model is performed, and equilibrium points are determined. The model has three equilibria, namely, the disease-free equilibrium, the bacteria-free equilibrium, and the endemic equilibrium. We use Castillo-Chavez theorem to establish the global stability of the disease-free equilibrium when the basic reproduction number is less than 1. The local asymptotic stability of the bacteria-free, and endemic equilibria are also established using the sign of the eigenvalues of the Jacobian matrix. We use the non-standard finite difference scheme and carried numerical simulations to confirm the theoretical results. We further show the impact of specific parameters on the dynamics of infectious individuals and observe that intervention is required in all the sub-populations by reducing the contact rate and vertical transmission to mininmize the number of infectious.

A novel indicator in epidemic monitoring through a case study of Ebola in West Africa (2014-2016).

The E/S (exposed/susceptible) ratio is analyzed in the SEIR model. The ratio plays a key role in understanding epidemic dynamics during the 2014-2016 Ebola outbreak in Sierra Leone and Guinea. The maximum value of the ratio occurs immediately before or after the time-dependent reproduction number (Rt) equals 1, depending on the initial susceptible population (S(0)). It is demonstrated that transmission rate curves corresponding to various incubation periods intersect at a single point referred to as the Cross Point (CP). At this point, the E/S ratio reaches an extremum, signifying a critical shift in transmission dynamics and aligning with the time when Rt approaches 1. By plotting transmission rate curves, β(t), for any two arbitrary incubation periods and tracking their intersections, we can trace CP over time. CP serves as an indicator of epidemic status, especially when Rt is close to 1. It provides a practical means of monitoring epidemics without prior knowledge of the incubation period. Through a case study, we estimate the transmission rate and reproduction number, identifying CP and Rt = 1 while examining the E/S ratio across various values of S(0).

Two strains model of infectious diseases for mathematical analysis and simulations

ABSTRACT In this study, we study a two-strain nonlinear model for the transmission of COVID-19 with a vaccinated class. Here, it is remarkable that the model we consider contains two kinds of viruses known as Omicron and Delta variants denoted by A and B , respectively. Also, the uninfected population is denoted by S , the vaccinated class by V and the recovered individuals by R . In the presented study, we consider the proposed model under conformable fractional order derivatives. The fundamental reproductive number and equilibrium points are computed. Moreover, we determine the existence and uniqueness of the solution to the suggested model using fixed-point theory. Furthermore, we provide a suitable methodology by applying the Euler numerical method to calculate the approximate solution of each compartment of the proposed model. Additionally, using MATLAB-16, we simulate the given results graphically for a variety of fractional orders using some real values of the parameters and initial conditions.

MODELING THE SPREAD OF COVID-19 DISEASE WITH TIME DELAY IN PONTIANAK CITY

Coronavirus disease 2019 (COVID-19) is an infectious disease caused by a coronavirus originating from the city of Wuhan in 2019. This disease affects the respiratory system. The city of Pontianak has the highest population density in West Kalimantan. This density results in a higher spread of Covid-19. In this article, the spread of COVID-19 is formulated into a mathematical model, equilibrium points are sought, stability is analyzed, and a delay time is introduced to reduce the spread of COVID-19. The magnitude of the delay time given during quarantine complies with health protocols, which is between 2 – 14 days. This article aims to analyze the influence of the delay time in modeling the spread of Covid-19. The problem of COVID-19 spread is constructed into an SIQR model, with a sub-population of recovered individuals returning to the susceptible sub-population. The population is divided into four sub-populations: susceptible (S), Infected (I), Quarantined (Q), and Recovered (R). The parameters used include the natural birth rate ( ), the rate of susceptibility to infection ( ), the rate of infection under quarantine ( ), the recovery rate from infection ( ), the recovery rate from infection under quarantine ( ), the death rate from infection ( ), the death rate under quarantine ( ), the delay time from infection to quarantine process ( ), the natural death rate ( ), and the rate of recovered immunity returning to susceptibility ( ). The simulation results show that when the basic reproduction number is less than , the disease-free equilibrium is stable, and when the basic reproduction number is greater than , the endemic equilibrium point is stable. The addition of a time delay ( ) in the SIQR model affects the stability of the endemic equilibrium point but does not affect the stability of the disease-free equilibrium point.

Analysis of a fractional-order model for dengue transmission dynamics with quarantine and vaccination measures

A comprehensive mathematical model is proposed to study two strains of dengue virus with saturated incidence rates and quarantine measures. Imperfect dengue vaccination is also assumed in the model. Existence, uniqueness and stability of the proposed model are proved using the results from fixed point and degree theory. Additionally, well constructed Lyapunov function candidates are also applied to prove the global stability of infection-free equilibria. It is also demonstrated that the model is generalized Ulam-Hyers stable under some appropriate conditions. The model is fitted to the real data of dengue epidemic taken from the city of Espirito Santo in Brazil. For the approximate solution of the model, a non-standard finite difference(NSFD) approach is applied. Sensitivity analysis is also carried out to show the influence of different parameters involved in the model. The behaviour of the NSFD is also assessed under different denominator functions and it is observed that the choice of the denominator function could influence the solution trajectories. Different scenario analysis are also assessed when the reproduction number is below or above one. Furthermore, simulations are also presented to assess the epidemiological impact of dengue vaccination and quarantine measures for infected individuals.

A stochastic model for the early stages of highly contagious epidemics by using a state-dependent point process

The recent COVID-19 pandemic has shown that when the reproduction number is high and there are no proper measurements in place, the number of infected people can increase dramatically in a short time, producing a phenomenon that many stochastic SIR-like models cannot describe: overdispersion of the number of infected people (i.e., the variance of the number of infected people during any interval is very high compared to the average). To address this issue, in this paper we explore the possibility of modeling the total number of infections as a state-dependent self-exciting point process. In this way, infections are not independent among themselves, but any infection will increase the likelihood of a new infection while also the number of currently infected and recovered individuals are included into determining the likelihood of new infections, Since long term simulation is extremely computationally intensive, exact expressions for the moments of the processes determining the number of infected and recovered individuals are computed, while also simulation algorithms for these state-dependent processes are provided.

Mathematical modeling of chickenpox transmission using the Laplace Adomian Decomposition Method

In this work, mathematical modeling of a nonlinear differential equation was studied to investigate the effect of vaccination on the spread of chickenpox. The proof of existence and uniqueness of the positive solution and invariant region showed that the model is epidemiologically sound. We established the disease-free and endemic equilibrium states and carried out a stability analysis of the disease-free and endemic equilibrium states of the model to gain insight into the dynamics of the model. The rate of vaccination and precaution for the spread of chickenpox was a factor that influenced the basic reproductive number, which was calculated using the next-generation matrix approach. Forecasts made via numerical simulation using the Laplace Adomian Decomposition method highlight the temporal impact of vaccination on curbing the chicken pox trend.

An Efficient Approach to Nowcasting the Time-varying Reproduction Number.

Estimating the instantaneous reproduction number () in near real time is crucial for monitoring and responding to epidemic outbreaks on a daily basis. However, such estimates often suffer from bias due to reporting delays inherent in surveillance systems. We propose a fast and flexible Bayesian methodology to overcome this challenge by estimating while taking into account reporting delays. Furthermore, the method naturally takes into account the uncertainty associated with the nowcasting of cases to get a valid uncertainty estimation of the nowcasted reproduction number. We evaluate the proposed methodology through a simulation study and apply it to COVID-19 incidence data in Belgium.

Age-Stage, Two-Sex Life Table of an Alarming Quarantine Pest Trogoderma granarium on Stored Products

Trogoderma granarium (Coleoptera: Dermestidae) is a polyphagous pest of stored products and an alarming threat to food security around the globe. The current study is designed to construct the life table of T. granarium on four stored products, i.e., wheat, barley, rice, and maize at controlled conditions (33±1°C, 65% RH, and a photoperiod of 14 : 10 (L:D) h). The incubation period of T. granarium was longer (5.13 d) on maize and shorter (4.30 d) on wheat. The developmental period of 1st, 2nd, 3rd, and 4th larval instars of T. granarium was longer on maize than the other three hosts were significant. Statistical difference between the longevity of females and males was recorded, while no statistically significant difference was found in the preoviposition period. The mean fecundity values observed on wheat, barley, rice, and maize were 31.33±1.62, 26.21±1.49, 28.00±1.35, and 25.58±1.13, respectively. The highest fertility was observed in wheat and the lowest in maize. The net reproductive rate (R0) of T. granarium ranged from 8.66±2.86, 13.16±2.65, 12.53±2.27, and 10.23±2.32 offsprings on wheat, barley, rice, and maize, respectively. The highest gross reproductive value (GRR) was for T. granarium reared on barley (18.31±4.10/d), and the lowest was for maize (15.35±2.95/d). To complete the development, T. granarium took 48.97±0.68 d on maize, significantly higher than on other treatments (p<0.05). The mean generation time (T) of T. granarium was considerably affected by tested products. Based on the results of the present study, maize is a less favorable diet for the development and reproduction of T. granarium and may be considered for the management of this detrimental pest.

Some novel results of a two-community SIR model with asymmetric structure

In this paper, a two-community SIR model with an asymmetric structure is proposed. We provide a concise method to derive an explicit formula of the basic reproduction number ℛ0 and present a route conception for the explanation of ℛ0. Finally, we find a close relationship between ℛ0 and the final size by the Perron–Frobenius theorem and the Krasnoselskii fixed point theorem.

Combining prior and post‐release data while accounting for dispersal to improve predictions for reintroduction populations

AbstractAttempts to reintroduce species to managed areas may be compromised by dispersal into the surrounding landscape. Therefore, decisions regarding the selection and ongoing management of reintroduction areas require predicting dispersal as well as the survival and reproduction rates of the species to be reintroduced. Dispersal can potentially be measured directly by tracking animals, but this is often impractical. However, dispersal can also be inferred from re‐sighting surveys done within reintroduction areas if such data are available from multiple areas with varying connectivity to the surrounding landscape, allowing apparent survival and recruitment to be modelled as a function of connectivity metrics. Here, we show how data from 10 previous reintroductions of a New Zealand passerine, the toutouwai (Petroica longipes), were used to predict population dynamics at a predator‐controlled reintroduction area with high connectivity, and predictions then updated using post‐release data. Bayesian hierarchical modelling of the previous data produced prior distributions for productivity, adult survival and apparent juvenile survival rates that accounted for random variation among areas as well as rat density and connectivity. The modelling of apparent juvenile survival as a function of connectivity allowed it to be partitioned into estimates of survival and fidelity. Bayesian updating based on post‐release data produced posterior distributions for parameters that were consistent with the priors but much more precise. The prior data also allowed the recruitment rate estimated in the new area to be partitioned into separate estimates for productivity, juvenile survival and juvenile fidelity. Consequently, it was possible to not only estimate population growth under current management, but also predict the consequences of reducing the scale or intensity of predator control, facilitating adaptive management. The updated model could then be used to predict population growth as a function of the connectivity and predator control regime at proposed reintroduction areas while accounting for random variation among areas.

Modelling the Time-varying Transmission of Wild Poliovirus in Pakistan

Aims/Objectives: The efforts to eradicate the wild poliovirus since 1988 have successfully reduced its global prevalence by 99%. However, as of 2024, Pakistan and Afghanistan remain the only two endemic countries facing the virus transmission. This study employs a transmission dynamic model to understand the persistence of wild poliovirus type 1 (WPV1) in Pakistan. Study Design: An ordinary differential equations-based deterministic type of mathematical model was developed. Place and Duration of Study: Department of Epidemiology and Public Health and Institute of Microbiology, University of Agriculture Faisalabad, March 2023 to August 2023. Methodology: We included the number of reported polio cases and the proportion of missed children by supplementary immunization activities (SIAs) across the country from 2017-2022. The model considered both human-human and environment-human virus transmission through sewage contamination and these are represented by time-dependent transmission rate parameters. The parameter estimation was done by model fitting to the reported data of WPV1 cases. The model simulations then predicted the future polio infections in Pakistan. Results: Our analysis identified that the asymptomatic infectious population missed by the SIAs is the major contributor to disease persistence in the country. Moreover, the environment can contribute to virus transmission in areas with poor WASH infrastructure and under-immunized populations. The estimated basic reproduction number through the next-generation matrix method was 1.61 while the estimated effective reproduction number was 0.12. The model showed a better predictive ability than constant transmission rate models. The numerical simulations considering the reduction in the virus-shedding rate by the asymptomatic population resulted in an 85% reduction in future polio cases in Pakistan. Conclusion: The results suggest that endemic virus transmission will continue subject to the current higher vaccination coverage across the country. The model can be further utilized to guide eradication efforts for targeted allocation of preemptive measures.

A new co-infection model for HBV and HIV with vaccination and asymptomatic transmission using actual data from Taiwan

The co-infection of Human Immunodeficiency Virus (HIV) and Hepatitis B virus (HBV) poses a major threat to public health due to their combined negative impacts on health and increased risk of complications. A novel fractional mathematical model of the dynamics of co-infection between HBV and HIV for Taiwan is presented in this paper. Detailed analyses are conducted on the possible impact of HBV vaccination on the dynamics of HBV and HIV co-infection. The next-generation matrix technique is used to calculate the fundamental reproduction number R 0 = max{R 1, R 2}, where R 1 and R 2 are the reproduction numbers for HBV and HIV, respectively. The disease-free and endemic equilibria of the co-infection model are calculated. An extensive investigation is carried out to determine the local and global stability of the disease-free equilibrium point through Rough Hurtwiz criteria and the construction of Lyapunov function, respectively. We demonstrate that when R 1 < 1 < R 2, HBV infection is eradicated, but HIV remains prevalent. If R 2 < 1 < R 1, the opposite outcome occurs. The real data from 2000-2023 for Taiwan is used to fit the model. The fitting results show how effectively our model handles the data. In addition, numerical simulations are run for different scenarios to observe how the vaccine and fractional parameters changed the model state variables, as well as how the solutions behaved and how quickly they reached the model’s equilibrium points. According to the model’s numerical analysis, greater vaccination efforts against HBV have a positive effect on the propagation of co-infection.

Sudan virus disease super-spreading, Uganda, 2022

BackgroundOn 20 September 2022, Uganda declared its fifth Sudan virus disease (SVD) outbreak, culminating in 142 confirmed and 22 probable cases. The reproductive rate (R) of this outbreak was 1.25. We described persons who were exposed to the virus, became infected, and they led to the infection of an unusually high number of cases during the outbreak.MethodsIn this descriptive cross-sectional study, we defined a super-spreader person (SSP) as any person with real-time polymerase chain reaction (RT-PCR) confirmed SVD linked to the infection of ≥ 13 other persons (10-fold the outbreak R). We reviewed illness narratives for SSPs collected through interviews. Whole-genome sequencing was used to support epidemiologic linkages between cases.ResultsTwo SSPs (Patient A, a 33-year-old male, and Patient B, a 26-year-old male) were identified, and linked to the infection of one probable and 50 confirmed secondary cases. Both SSPs lived in the same parish and were likely infected by a single ill healthcare worker in early October while receiving healthcare. Both sought treatment at multiple health facilities, but neither was ever isolated at an Ebola Treatment Unit (ETU). In total, 18 secondary cases (17 confirmed, one probable), including three deaths (17%), were linked to Patient A; 33 secondary cases (all confirmed), including 14 (42%) deaths, were linked to Patient B. Secondary cases linked to Patient A included family members, neighbours, and contacts at health facilities, including healthcare workers. Those linked to Patient B included healthcare workers, friends, and family members who interacted with him throughout his illness, prayed over him while he was nearing death, or exhumed his body. Intensive community engagement and awareness-building were initiated based on narratives collected about patients A and B; 49 (96%) of the secondary cases were isolated in an ETU, a median of three days after onset. Only nine tertiary cases were linked to the 51 secondary cases. Sequencing suggested plausible direct transmission from the SSPs to 37 of 39 secondary cases with sequence data.ConclusionExtended time in the community while ill, social interactions, cross-district travel for treatment, and religious practices contributed to SVD super-spreading. Intensive community engagement and awareness may have reduced the number of tertiary infections. Intensive follow-up of contacts of case-patients may help reduce the impact of super-spreading events.

Distribution patterns of breeding nests in the colonial-breeding Azure-winged Magpie Cyanopica cyanus

ABSTRACT Capsule Distribution patterns of the breeding nests of the Azure-winged Magpie can help to understand spatial and temporal changes in core use areas throughout the breeding range. Aims We investigated the spatial and temporal patterns of nest locations in the breeding colony, to determine the effect of breeding experience and timing on the nest location of Azure-winged Magpies. Methods We collected data on the nest locations of Azure-winged Magpies in a breeding colony and analysed them to identify patterns in nest location. We estimated the nest distribution range through a kernel density analysis, using a geographic information system. Results The size and location of the core use area of the breeding population changed in the following breeding season when the previous year's reproductive success rate was low. We also observed changes in the size and location of the core-use area in relation to the timing of breeding. Conclusion Inter-annual change of the core use area was associated with the timing of breeding and the population's breeding experience.

Taylor’s law and quasi-experimental hunting of invasive wild boar and axis deer in a protected area of north-eastern Argentina

Context Overabundant ungulate populations frequently cause ecological and economic damage and are difficult to control. Sustained culling efforts at El Palmar National Park have largely reduced wild boar (Sus scrofa) numbers and damage but have failed to halt the ever-increasing abundance of axis deer (Axis axis) for undefined reasons. Multi-year camera-trap surveys indicated a mismatch between deer core activity and the usual timing of hunting sessions. Aim We used the spatial and temporal forms of Taylor’s law (TL, a power relationship between the sample mean and sample variance) to test whether overnight hunting sessions are more effective for culling axis deer and wild boar than are daytime sessions, and assessed whether they modify ungulate harvest composition. Methods We implemented a quasi-experimental trial (i.e. ‘experiments in which units are not assigned to conditions randomly’), including nine alternating blocks of four to seven hunting sessions each (overnight or daytime) in which an average of 42 hunter groups performed controlled shooting over bait from fixed elevated blinds, retrieved, sexed, weighed and measured all quarry. Key results The relationship between the mean and variance of ungulate crude catch per hunting-party session and crude catch per unit effort (CPUE, where effort is measured in hunting-party hours) by hunting shift satisfied the spatial and temporal TL. On average, axis deer catch was 2.39–2.61 times greater in overnight than in daytime sessions, whereas CPUE indices were 1.54–1.73 times greater. For wild boar, overnight sessions returned catches similar to (0.94–1.03×), and a significantly lower CPUE (0.63–0.67×) than for daytime sessions. The harvest indices were substantially aggregated in daytime sessions only, and were consistently skewed towards yearlings or subadults and males, especially among adult deer. Overnight sessions culled proportionally more adults and more pregnant hinds and sows than did daytime sessions, and significantly heavier deer and wild boar males. Conclusions Overnight sessions were substantially more effective for culling deer than were daytime sessions, and selected for individuals with greater fitness. Stage- and male-skewed harvest most likely explains the steady population growth of axis deer despite increasing hunting pressure over a decade. Implications Adjusting the schedule of hunting sessions to time intervals of peak ungulate activity substantially increased the efficiency of management efforts in terms of removal rates and reproductive value of the culled specimens.

A holistic exploration of the optimal control strategies on an enhanced mathematical model for the co-infection of HIV/AIDS and varicella-zoster

Because of its high contagiousness and correlation with HIV/AIDS complaints, the virus that causes varicella-zoster virus and its interactions have major consequences for a considerable portion of people worldwide. The primary aim of this work is to suggest and examine optimal control methods for managing the transmission dynamics of HIV/AIDS and Varicella-Zoster co-infection, using an integer model approach. The mathematical analyses of the proposed integer order model places particular emphases on the boundedness and non-negativity of the model solutions, scrutinizing equilibrium points, determining the models basic reproduction ratios (the models basic reproduction numbers) through the next-generation matrix operator method, and assessing the model equilibrium points existences and stabilities in local approach by considering the local stability conditions of Routh and Hurwitz. Additionally, it incorporates an optimal control framework to enhance our understanding of the dynamics involved in the spreading of HIV/AIDS and Varicella-Zoster co-infection within a considered population. This entails determining preventative measures that can be deliberately put into place to lessen the effects of these co-infections. The solutions of the HIV/AIDS and Varicella-Zoster co-infection model converges to the co-infection endemic equilibrium point whenever the associated basic reproduction number is greater than unity, as verified by numerical simulation results. Including optimal management gives the research an innovative viewpoint and helps identify tactical ways to mitigate the negative effects of this co-infection on the public health. The results highlight how crucial it is to address these complex structures in order to protect and improve public health outcomes. Implementing the proposed protection measures and treatment measures simultaneously has most effective result to minimize and eliminate the HIV/AIDS and Varicella-Zoster co-infection disease throughout the population.

Modeling and stability analysis of the transmission dynamics of Monkeypox with control intervention

Monkeypox is a deadly disease from the Orthopox family. The paper explores a nonlinear trend in the dynamics of Monkeypox propagation. In this model, we include vaccination, treatment, and level of awareness as the main parameters of the propagation of Monkeypox. We also include the quarantined infected individuals with severe complications in the Monkeypox dynamics. We show the boundedness and positivity of the model and also demonstrate the stability of the model using the Banach fixed point theory and the Picard successive approximation method to ensure that it is meaningful mathematically and epidemiologically. We obtain the unique solution of the model under appropriate conditions. This work finds three equilibrium points: Monkeypox free, Rodents free, and endemic. We show the local stability of monkeypox free equilibria and rodent-free endemic equilibria through reproduction number. Furthermore, Dulac’s function technique demonstrates the global stability of endemic equilibrium. We show the transformation from endemic equilibria to Monkeypox-free equilibria via rodent-free. We also obtain the condition of transcritical bifurcation. We analyze the system’s dynamic behavior to develop efficient infection control strategies. We find that if we increase awareness, vaccination, treatment, and quarantine rates, we effectively control the transmission. The model is connected with a continuous model using an ordinary differential equation. We examine the complex dynamics of Monkeypox infection under diverse system input factors through numerical simulation of the proposed model with variable input parameters in reducing Monkeypox. We present the theory of Monkeypox disease control in humankind as an application in real-world problems. The work can be helpful in the vector-borne disease control system.

Dual-strain dynamics of COVID-19 variants in India: Modeling, analysis, and implications for pandemic control

This study introduces a detailed compartmental model developed to understand the complex dynamics of COVID-19 transmission, focusing on the Delta and Omicron variants in India. The model tracks disease progression through different population compartments, considering factors like vaccination, time-dependent transmission, economic burden and COVID-19 death rates, loss of vaccine-induced immunity, and the transition of asymptomatic cases to recovery. The model is validated against established epidemiological knowledge and real-world data, emphasizing dynamic parameterization and accurate representation of immunity dynamics. The basic reproduction number for both variants is calculated, and sensitivity analysis for various parameters is conducted. Time-dependent parameters are estimated using the discrete inverse method. The study also explores the economic burden, impact of different types of masks, vaccine efficacy, and vaccine-induced immunity through numerical analysis.

A taste of space: Remote animal observations and discrete-choice models provide new insights into foraging and density dynamics for a large subarctic herbivore.

Competition for resources and space can drive forage selection of large herbivores from the bite through the landscape scale. Animal behaviour and foraging patterns are also influenced by abiotic and biotic factors. Fine-scale mechanisms of density-dependent foraging at the bite scale are likely consistent with density-dependent behavioural patterns observed at broader scales, but few studies have directly tested this assertion. Here, we tested if space use intensity, a proxy of spatiotemporal density, affects foraging mechanisms at fine spatial scales similarly to density-dependent effects observed at broader scales in caribou. We specifically assessed how behavioural choices are affected by space use intensity and environmental processes using behavioural state and forage selection data from caribou (Rangifer tarandus granti) observed from GPS video-camera collars using a multivariate discrete-choice modelling framework. We found that the probability of eating shrubs increased with increasing caribou space use intensity and cover of Salix spp. shrubs, whereas the probability of eating lichen decreased. Insects also affected fine-scale foraging behaviour by reducing the overall probability of eating. Strong eastward winds mitigated negative effects of insects and resulted in higher probabilities of eating lichen. At last, caribou exhibited foraging functional responses wherein their probability of selecting each food type increased as the availability (% cover) of that food increased. Space use intensity signals of fine-scale foraging were consistent with density-dependent responses observed at larger scales and with recent evidence suggesting declining reproductive rates in the same caribou population. Our results highlight potential risks of overgrazing on sensitive forage species such as lichen. Remote investigation of the functional responses of foraging behaviours provides exciting future applications where spatial models can identify high-quality habitats for conservation.